INMO–2006

1. In a nonequilateral triangle ABC, the sides a, b, c form an arithmeticprogression. Let I and O denote the incentre and circumcentre of thetriangle respectively.

(i) Prove that IO is perpendicular to BI.

(ii) Suppose BI extended meets AC in K, and D,E are the mid-points of BC, BA respectively. Prove that I is the circumcentreof triangle DKE.

2. Prove that for every positive integer n there exists a unique orderedpair (a, b) of positive integers such that

n =12(a + b− 1)(a + b− 2) + a.

3. Let X denote the set of all triples (a, b, c) of integers. Define a functionf : X → X by

f(a, b, c) = (a + b + c, ab + bc + ca, abc).

Find all triples (a, b, c) in X such that f(f(a, b, c)) = (a, b, c).

4. Some 46 squares are randomly chosen from a 9 × 9 chess board andare coloured red. Show that there exists a 2× 2 block of 4 squares ofwhich at least three are coloured red.

5. In a cyclic quadrilateral ABCD, AB = a, BC = b, CD = c, ∠ABC =120◦, and ∠ABD = 30◦. Prove that

(i) c ≥ a + b;

(ii) |√

c + a−√

c + b| =√

c− a− b.

6. (a) Prove that if n is a positive integer such that n ≥ 40112, thenthere exists an integer l such that n < l2 < (1 + 1

2005)n.

(b) Find the smallest positive integer M for which whenever an in-teger n is such that n ≥ M , there exists an integer l, such thatn < l2 < (1 + 1

2005)n.